Already in 1962-63 in SGA4 was considered and developed the concept of Fibred Topos (SLN 270 VI 7. p 273). Recall that in SGA4, Topos (or U-topos) = your Bounded Topos. If you want to consider a different concept of fibred topos, it is not correct to use the same name. best eduardo On 18/12/12 03:28, David Roberts wrote:
Dear all,
I'm thinking about fibred toposes, and I was wondering if there any references people can suggest? The following are some pitifully vague thoughts.
One particular problem I'm thinking about is whether there is a generic fibred topos, which is the analogue of the generic discrete fibration Set_* --> Set or the generic fibration 1 / Cat --> Cat.
Something like the 2-category Topos of bounded toposes and geometric morphisms (and whatever 2-arrows are appropriate). The objects of this are bounded geometric morphisms, arrows are 2-commutative squares. Then take the 2-category over this where the objects are bounded toposes E --> S with a point Set --> E, or possibly an S-point S --> E, and arrows those geometric morphisms which preserve the point up to natural transformation.
Ideally I'd then like to consider 2-functors T^op -->Topos to be equivalent to (bounded) fibred toposes over T.
Best,
David Roberts
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